A190617 The smallest prime q <= prime(n) such that 1 + q# * prime(n)# is prime, or 0 if no such q exists.

2, 2, 2, 2, 2, 3, 0, 19, 13, 13, 2, 11, 0, 3, 0, 7, 3, 2, 0, 0, 3, 0, 2, 0, 7, 2, 0, 0, 7, 2, 0, 0, 5, 13, 17, 5, 0, 29, 73, 53, 0, 41, 17, 0, 61, 113, 67, 0, 23, 7, 31, 53, 3, 0, 0, 109, 13, 43, 101, 67, 113, 0, 181, 37, 23

Offset

1,1

Comments

The notation # refers to the primorials A002110, the partial products of primes.

Roughly 75% of the entries are nonzero.

For roughly 50% of the solutions (that is roughly 1/3 of all entries if zeros are included) the q are smaller than prime(n)/5.

Links

Pierre CAMI, Table of n, a(n) for n = 1..750

Example

2*2 + 1 = 5 (prime) with q=2, q#=2, prime(n)# = 2 so a(1)=2.
2*2*3 + 1 = 13 (prime) with q=2, q#=2, prime(2)# = 2*3 so a(2)=2.
2*2*3*5 + 1 = 61 (prime) with q=2, q#=2, prime(3)# = 2*3*5 so a(3)=2.
2*2*3*5*7 + 1 = 421 (prime) with q=2, q#=2, prime(4)# = 2*3*5*7 so a(4)=2.

Maple

A002110 := proc(n) option remember; mul(ithprime(i), i=1..n) ; end proc:
A190617 := proc(n) local psharp ; psharp := A002110(n) ; for i from 1 to n do if isprime(1+psharp*A002110(i)) then return ithprime(i) ; end if; end do: return 0 ; end proc:
seq(A190617(n), n=1..80) ; # R. J. Mathar, Jun 02 2011

Prog

PFGW from Primeformgroup for prime search and certification
pfgw64 -f in.txt , results in pfgw-prime.log and pfgw.log
in.txt scriptyfile
SCRIPT
DIM nn, 0
DIM kk
DIM mm
DIM jj
DIMS tt
LABEL loopn
SET nn, nn+1
IF nn>750 THEN END
SET kk, p(nn)
SET mm, 0
LABEL loopm
SET mm, mm+1
IF mm>nn THEN GOTO loopn
SET jj, p(mm)
SETS tt, %d, %d\,; kk; jj
PRP kk#*(jj#)+1, tt
IF ISPRIME THEN GOTO loopn
  IF ISPRP THEN GOTO loopn
goto loopm

Crossrefs

Sequence in context: A139514 A330955 A330956 * A068323 A054990 A046921

Adjacent sequences: A190614 A190615 A190616 * A190618 A190619 A190620

Keyword

nonn

Author

Pierre CAMI, May 14 2011

Status

approved